Problem: Simplify the following expression: $r = \dfrac{-24x^2 - 48x}{54x^2}$ You can assume $x \neq 0$.
Solution: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-24x^2 - 48x = - (2\cdot2\cdot2\cdot3 \cdot x \cdot x) - (2\cdot2\cdot2\cdot2\cdot3 \cdot x)$ The denominator can be factored: $54x^2 = (2\cdot3\cdot3\cdot3 \cdot x \cdot x)$ The greatest common factor of all the terms is $6x$ Factoring out $6x$ gives us: $r = \dfrac{(6x)(-4x - 8)}{(6x)(9x)}$ Dividing both the numerator and denominator by $6x$ gives: $r = \dfrac{-4x - 8}{9x}$